Optimal. Leaf size=25 \[ \frac{i a (c-i c \tan (e+f x))^4}{4 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0676489, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3522, 3487, 32} \[ \frac{i a (c-i c \tan (e+f x))^4}{4 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx &=(a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^3 \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int (c+x)^3 \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac{i a (c-i c \tan (e+f x))^4}{4 f}\\ \end{align*}
Mathematica [B] time = 1.40167, size = 85, normalized size = 3.4 \[ \frac{a c^4 \sec (e) \sec ^4(e+f x) (2 \sin (e+2 f x)-2 \sin (3 e+2 f x)+\sin (3 e+4 f x)-2 i \cos (e+2 f x)-2 i \cos (3 e+2 f x)-3 \sin (e)-3 i \cos (e))}{4 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.003, size = 48, normalized size = 1.9 \begin{align*}{\frac{a{c}^{4} \left ( \tan \left ( fx+e \right ) +{\frac{i}{4}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}- \left ( \tan \left ( fx+e \right ) \right ) ^{3}-{\frac{3\,i}{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.68881, size = 81, normalized size = 3.24 \begin{align*} \frac{i \, a c^{4} \tan \left (f x + e\right )^{4} - 4 \, a c^{4} \tan \left (f x + e\right )^{3} - 6 i \, a c^{4} \tan \left (f x + e\right )^{2} + 4 \, a c^{4} \tan \left (f x + e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.33357, size = 158, normalized size = 6.32 \begin{align*} \frac{4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.26796, size = 83, normalized size = 3.32 \begin{align*} \frac{4 i a c^{4} e^{- 8 i e}}{f \left (e^{8 i f x} + 4 e^{- 2 i e} e^{6 i f x} + 6 e^{- 4 i e} e^{4 i f x} + 4 e^{- 6 i e} e^{2 i f x} + e^{- 8 i e}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.68332, size = 82, normalized size = 3.28 \begin{align*} \frac{4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]